Characteristic state function

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Microcanonical_ensemble/L4fE6SxT">microcanonical ensemble</a> satisfies 
  
    
      
        Ω
        (
        U
        ,
        V
        ,
        N
        )
        =
        
          e
          
            β
            T
            S
          
        
        
        
      
    
    {\displaystyle \Omega (U,V,N)=e^{\beta TS}\;\,}
  
 hence, its characteristic state function is 
  
    
      
        T
        S
      
    
    {\displaystyle TS}
  
.</li>
<li>The <a href="/facts/Canonical_ensemble/GIhyzkb8">canonical ensemble</a> satisfies 
  
    
      
        Z
        (
        T
        ,
        V
        ,
        N
        )
        =
        
          e
          
            −
            β
            A
          
        
        
        
      
    
    {\displaystyle Z(T,V,N)=e^{-\beta A}\,\;}
  
 hence, its characteristic state function is the <a href="/facts/Helmholtz_free_energy/2AK0HpPe">Helmholtz free energy</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
.</li>
<li>The <a href="/facts/Grand_canonical_ensemble/cvztr5Y0">grand canonical ensemble</a> satisfies 
  
    
      
        
          
            Z
          
        
        (
        T
        ,
        V
        ,
        μ
        )
        =
        
          e
          
            −
            β
            Φ
          
        
        
        
      
    
    {\displaystyle {\mathcal {Z}}(T,V,\mu )=e^{-\beta \Phi }\,\;}
  
, so its characteristic state function is the <a href="/facts/Grand_potential/oWRmLQSK">Grand potential</a> 
  
    
      
        Φ
      
    
    {\displaystyle \Phi }
  
.</li>
<li>The <a href="/facts/Isothermal-isobaric_ensemble/pFcQL5hQ">isothermal-isobaric ensemble</a> satisfies 
  
    
      
        Δ
        (
        N
        ,
        T
        ,
        P
        )
        =
        
          e
          
            −
            β
            G
          
        
        
        
      
    
    {\displaystyle \Delta (N,T,P)=e^{-\beta G}\;\,}
  
 so its characteristic function is the <a href="/facts/Gibbs_free_energy/bvenDuzh">Gibbs free energy</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
.</li></ul>
<p>State functions are those which tell about the equilibrium state of a system
</p>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Balian, Roger (2017-11-01). "François Massieu and the thermodynamic potentials". Comptes Rendus Physique. 18 (9–10): 526–530. Bibcode:2017CRPhy..18..526B. doi:10.1016/j.crhy.2017.09.011. ISSN 1631-0705. "Massieu's potentials [...] are directly recovered as logarithms of partition functions." <a href="https://doi.org/10.1016%2Fj.crhy.2017.09.011" target="_blank">https://doi.org/10.1016%2Fj.crhy.2017.09.011</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Characteristic state function open-in-new

Characteristic state function